I still don't see the issue. Ingenic is a Chinese SoC manufacturer making products based on the MIPS architecture. Intel might like to sell them Atom based SoCs, but that might require changes to Ingenic's strategic plan and business model. One question is whether Ingenic has its own fab and makes its own chips. What would they buy from Intel if they do? Do they close their own foundry and buy parts from Intel, do they license Intel IP to make in their own foundry, or do they keep their own foundry making MIPS chips and buy Atom chips from Intel?
And part of the confusion may lie in what constitutes the product. ARM and MIPS are fabless semi-conductor firms. They license designs that other vendors will actually make chips using. For ARM and MIPS, a design win is a license by a manufacturer to use their IP, with a royalty stream based on production.
Intel is a fabbed semi-conductor manufacturer. They both make designs and manufacture chips using those designs. So for Intel, a design win might be either a license of IP with royalties based on production, or an order for X thousands of physical chips to build into someone else's device. Which Intel might prefer to get will probably depend upon estimated volume.
But if Intel buys MIPS, I *don't* see them canceling existing licensing agreements. Not only would it be difficult to do: they'd be buying MIPS in part to *get* those customers. Canceling existing agreements would require OEMs to renegotiate to continue production, and they might just switch to ARM. It would probably also make OEMs who *hadn't* previously licensed MIPS IP reluctant to do so, as the question would be "Can we trust Intel not to screw us?", and the answer might be "On evidence, no."
If I'm Intel and I buy MIPS to get a piece of the mobile market, I keep existing licenses in place, and what I sell depends on the market. I'm not going to convert one of my fabs to MIPS production unless I have a really big design win to pay for it.

Perhaps a new Chinese company is created to buy MIPS. It converts the architecture into open source and becomes a design services company. China rallies behind MIPS and the ecosystem is strengthened because like Linux its free. Big picture, China wins since it spends less on foreign ARM/Intel IP.
??

I'm curious as to why you wouldn't discount the patent portfolio. It does seem very reasonable to ask ... if MIPS is not successfully asserting patents rights against ARM/Intel/AMD now, why would someone be able to in the future or how would the patents otherwise be useful?

Perhaps explain myself well. My meaning was that Intel licensing a MIPS core for use in an SoC by Ingenic would then be competing with Ingenic when it tries to sell an Atom-based SoC.
Of course, if Intel buys MIPS and stops all licensing operations and offers Atom- or MIPS-based SoCs there is no competition. But terminating those licensing agreements is not necessarily easily done.

'BUT I still feel they would fall foul of the "don't compete with your customers" rule.'
I don't see how Intel buying MIPS and offering MIPS cores falls under this rule. Intel's customers are OEMs using chips. Was Intel competing with their customers when they licensed ARM IP and sold ARM CPUs through their former StrongARM division? Why should acquiring MIPS and offering their cores instead be any different?
"How could Intel license high-end and mobile MIPS cores when it is trying to sell Atom and Ivy Bridge into the same sectors?"
That would be a matter of competing with themselves. For the customer, it's an advantage: is what they are trying to do better served by Atom and Ivy Bridge, or by MIPS designs? And it's an advantage for Intel, too, as it broadens their product offerings and gives them more options in competing.
You might get the sort of fun I observed decades back, when IBM salesmen for the AS400 and RS6000 lines danced around each other, as there was overlap in what those products could be used for, and two IBM divisions might be competing head to head for the same customer.

My bet in this speculative discussion is on Intel too. They desperately need an out into the mobile market; and with ARM infringing on the server market it makes sednse for Intel to use its cash to do with MIPS as they see fit. The question is, is there a tight fit between MIPS and Intel strategies? For a company that capitalized on a radically different RISC architecture developed by current Stanford U. president J. Hennessy MIPS might be just what Intel needs to counter the ARM-based mobile juggernaut.

With China's recent commitment to a national silicon standard (ISA), a Chinese firm (newly created, established, or government) makes perfect sense as a purchaser. I suspect that any regulatory and security concerns can be negotiated.